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We investigate a primal-dual (PD) method for the saddle point problem (SPP)that uses a linear approximation of the primal function instead of the standardproximal step, resulting in a linearized PD (LPD) method. For convex-stronglyconcave SPP, we observe that the LPD method has a suboptimal dependence on theLipschitz constant of the primal function. To fix this issue, we combinefeatures of Accelerated Gradient Descent with the LPD method resulting in asingle-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD methodachieves the optimal gradient complexity when the SPP has a semi-linearcoupling function. We also present an inexact ALPD method for SPPs with ageneral nonlinear coupling function that maintains the optimal gradientevaluations of the primal parts and significantly improves the gradientevaluations of the coupling term compared to the ALPD method. We verify ourfindings with numerical experiments.

Accelerated Primal-Dual Methods for Convex-Strongly-Concave Saddle Point Problems